For a Banach function lattice \(X\) , if all equimeasurable functions \(f,g \in X\)  satisfies \(\Vert f\Vert _X = \Vert g\Vert _X\) , then we say \(X\)  is rearrangement invariant (r.i.). Orlicz-Lorentz function and sequence spaces and their Köthe dual spaces are examples of r.i. Banach function and sequence spaces. A fundamental function is defined by \(\phi _X(t)= \Vert \chi _{E_t}\Vert _X\)  where \(E_t\)  is a set of measure \(t \in [0, \infty ]\) . For more details on the theory of Banach function lattices, we refer to [1]}, [2]}, [3]}.
